Finite Group Databasebeta

← C2×#20246064 #26375014

Subgroups — C2×#20246064 #26375014

Order 12032.

3198 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094000C211yesmaximal, characteristic
2C22×#13093981C211yesmaximal, characteristic
3#20246064C2116yesmaximal, direct product factor
4C2×#15689030C211yesmaximal, characteristic, complemented
5C2×#15689019C211yesmaximal, characteristic, complemented
6C2×#15689018C212yesmaximal, complemented
7C2×#15689014C212yesmaximal
8C2×#15689013C212yesmaximal
9C2×#15689007C212yesmaximal, complemented
10C2×#15688980C211yesmaximal, characteristic
11C2×C4×C4⋊Dic47C211yesmaximal, characteristic, complemented
12C22×C4×Q8×C47C211yesmaximal, characteristic, complemented
13C23×C4×D47C2211yescharacteristic
14C2×#13094000C4116yescomplemented
15C2×#13094000C2218yes—
16C2×#13093981—28no—
17C2×#13093981C2218yes—
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20C22×Dic47⋊C4C2211yescharacteristic
21C22×Dic47⋊C4C2211yescharacteristic
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×(C2×D47)⋊C4C2211yescharacteristic
25C22×(C2×D47)⋊C4C2211yescharacteristic
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×C4×Dic47C2211yescharacteristic
29C22×C4×Dic47C2212yes—
30#15689030C2218yescomplemented
31#15689019C2218yescomplemented
32#15689018C22116yescomplemented
33#15689014C22116yes—
34#15689013C22116yes—
35#15689007C22116yescomplemented
36#15688980C2218yes—
37C4×C4⋊Dic47C2218yescomplemented
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C23×Q8×C47C2211yescharacteristic
44C2×C4×Q8×C47—28no—
45C2×C4×Q8×C47C2218yescomplemented
46C22×C4⋊C4×C47C2212yes—
47C22×C4⋊C4×C47C2211yescharacteristic
48C22×C42×C47C2212yes—
49C22×C42×C47C2211yescharacteristic
50C24×D47C2311yescharacteristic
51C23×Dic47C2311yescharacteristic
52C23×Dic47C2312yes—
53C23×Dic47C2311yescharacteristic
54C23×Dic47C2312yes—
55C23×Dic47C2311yescharacteristic
56C22×C4×D47—24no—
57C22×C4×D47—24no—
58C22×C4×D47C2×C418yes—
59C22×C4×D47C2314yes—
60#13094000C2×C4164yescomplemented
61#13093981—232no—
62C2×C4⋊Dic47—24no—
63C2×C4⋊Dic47C2×C418yes—
64C2×C4⋊Dic47C2314yes—
65C2×C4⋊Dic47C2314yes—
66C2×Dic47⋊C4—24no—
67C2×Dic47⋊C4—24no—
68C2×Dic47⋊C4—44no—
69C2×Dic47⋊C4C2×C4116yes—
70C2×Dic47⋊C4C2314yes—
71C2×Dic47⋊C4C2314yes—
72C2×Dic47⋊C4C2318yes—
73C2×Dic47⋊C4C2318yes—
74C2×(C2×D47)⋊C4—24no—
75C2×(C2×D47)⋊C4—24no—
76C2×(C2×D47)⋊C4—44no—
77C2×(C2×D47)⋊C4C2×C4116yes—
78C2×(C2×D47)⋊C4C2314yes—
79C2×(C2×D47)⋊C4C2314yes—
80C2×(C2×D47)⋊C4C2318yes—
81C2×(C2×D47)⋊C4C2318yes—
82C2×C4×Dic47—24no—
83C2×C4×Dic47—44no—
84C2×C4×Dic47C2314yes—
85C2×C4×Dic47C2318yes—
86C47⋊#95C2318yes—
87C47⋊#95C2318yes—
88C47⋊#95C2318yes—
89C47⋊#95C2318yes—
90C47⋊#95C2318yes—
91C22×Q8×C47—24no—
92C22×Q8×C47—24no—
93C22×Q8×C47C2×C418yes—
94C22×Q8×C47C2314yes—
95C23×C4×C47C2312yes—
96C23×C4×C47C2312yes—
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2311yescharacteristic
100C4×Q8×C47—232no—
101C2×C4⋊C4×C47—28no—
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47C2314yes—
104C2×C4⋊C4×C47C2318yes—
105C2×C42×C47—28no—
106C2×C42×C47D418yes—
107C2×C42×C47C2318yes—
108C2×C42×C47C2314yes—
109C24×C47C2411yescharacteristic
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—22no—
116C23×D47C22×C414yes—
117C23×D47C2412yes—
118C2×Q8×C47—232no—
119C2×Q8×C47—28no—
120C2×Q8×C47—28no—
121C2×Q8×C47C22×C4116yes—
122C22×C4×C47—24no—
123C22×C4×C47—24no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47C22×C418yes—
132C22×C4×C47C22×C414yes—
133C22×C4×C47C2×D414yes—
134C22×C4×C47C2×D414yes—
135C22×C4×C47C2×D414yes—
136C22×C4×C47#4114yes—
137C22×C4×C47C2414yes—
138C22×C4×C47C2414yes—
139C22×C4×C47C2412yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2412yes—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—24no—
145C22×Dic47—24no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—24no—
155C22×Dic47C22×C418yes—
156C22×Dic47C22×C414yes—
157C22×Dic47C2412yes—
158C22×Dic47C2414yes—
159C22×Dic47C2412yes—
160C22×Dic47C2414yes—
161C22×Dic47C2412yes—
162C2×C4×D47—232no—
163C2×C4×D47—28no—
164C2×C4×D47—28no—
165C2×C4×D47C22×C4116yes—
166C4⋊C4×C47—216no—
167C4⋊C4×C47—28no—
168C42×C47—216no—
169C42×C47C2×D4116yes—
170(C2×D47)⋊C4—28no—
171(C2×D47)⋊C4—28no—
172(C2×D47)⋊C4—48no—
173(C2×D47)⋊C4C22×C4132yes—
174C4⋊Dic47—28no—
175C4⋊Dic47C22×C4116yes—
176Dic47⋊C4—28no—
177Dic47⋊C4—28no—
178Dic47⋊C4—48no—
179Dic47⋊C4C22×C4132yes—
180C4×Dic47—28no—
181C4×Dic47—48no—
182C23×C47C23×C412yes—
183C23×C47C22×D412yes—
184C23×C47C2×#4112yes—
185C23×C47H(5,𝔽2)12yes—
186C23×C472−1+412yes—
187C23×C472−1+412yes—
188C23×C472−1+412yes—
189C23×C47C2511yescharacteristic
190C22×D47—28no—
191C22×D47—28no—
192C22×D47—28no—
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—44no—
200C22×D47—44no—
201C22×D47—44no—
202C22×D47—44no—
203C22×D47C23×C414yes—
204Q8×C47—232no—
205C2×C4×C47—216no—
206C2×C4×C47—28no—
207C2×C4×C47—28no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—48no—
218C2×C4×C47C4×D4116yes—
219C2×C4×C47C23×C418yes—
220C2×C4×C47C23×C414yes—
221C2×C4×C47C22×D414yes—
222C2×C4×C47C22×D414yes—
223C2×C4×C47C22×D414yes—
224C2×C4×C47C2×#4114yes—
225C2×Dic47—28no—
226C2×Dic47—216no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—44no—
241C2×Dic47—44no—
242C2×Dic47—48no—
243C2×Dic47C23×C414yes—
244C2×Dic47C23×C418yes—
245C4×D47—232no—
246C2×#2323—471nomaximal, Sylow
247C22×C47C2×C4×D414yes—
248C22×C47#51914yes—
249C22×C47#52014yes—
250C22×C47#53614yes—
251C22×C47#53714yes—
252C22×C47#54114yes—
253C22×C47#54214yes—
254C22×C47C24×C411yescharacteristic
255C22×C47C23×D411yescharacteristic
256C22×C47C22×#4111yescharacteristic
257C22×C47C2×H(5,𝔽2)11yescharacteristic
258C22×C47C2×2−1+411yescharacteristic
259C22×C47C2×2−1+411yescharacteristic
260C22×C47C2×2−1+411yescharacteristic
261C2×D47—24no—
262C2×D47—24no—
263C2×D47—24no—
264C2×D47—416no—
265C2×D47—42no—
266C2×D47—42no—
267C2×D47—42no—
268C2×D47—42no—
269C4×C47—28no—
270C4×C47—24no—
271C4×C47—24no—
272C4×C47—44no—
273C4×C47C2×C4×D418yes—
274Dic47—24no—
275Dic47—28no—
276Dic47—42no—
277Dic47—42no—
278Dic47—44no—
279C22×C22⋊Q8—471no—
280C22×C4×Q8D4711yescharacteristic, complemented
281C22×#117—471no—
282#2323—4716no—
283C2×#391—471no—
284C2×#389—472no—
285C2×#389—471no—
286C2×#388—472no—
287C2×#388—472no—
288C2×#387—472no—
289C2×#380—471no—
290C2×C4×C4⋊C4—471no—
291C2×C47#232318yescomplemented
292C2×C47C22×C4×D411yescharacteristic
293C2×C47C2×#51911yescharacteristic
294C2×C47C2×#52011yescharacteristic
295C2×C47C2×#53611yescharacteristic
296C2×C47C2×#53711yescharacteristic
297C2×C47C2×#54111yescharacteristic
298C2×C47C2×#54211yescharacteristic
299D47—44no—
300C23×Q8C2×D4711yescharacteristic
301C24×C4—471no—
302C2×C22⋊Q8—4716no—
303C2×C22⋊Q8—478no—
304C2×C4×Q8—28no—
305C2×C4×Q8C2×D4718yescomplemented
306C2×#117—478no—
307C2×#117—948no—
308C22×C4⋊C4—472no—
309C22×C4⋊C4—471no—
310C22×C4⋊C4—471no—
311C22×C4⋊C4—471no—
312C22×C4⋊C4—471no—
313C22×C4⋊C4—472no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C4⋊C4C2×D4712yes—
316C22×C22⋊C4—472no—
317C22×C22⋊C4—471no—
318C22×C22⋊C4—471no—
319C22×C22⋊C4—472no—
320C22×C42—471no—
321C22×C42—472no—
322C22×C42C2×D4712yes—
323C22×C42C2×D4711yescharacteristic
324#391—478no—
325#389—4716no—
326#389—478no—
327#388—4716no—
328#388—4716no—
329#387—4716no—
330#380—478no—
331C4×C4⋊C4—478no—
332C2×#95—472no—
333C2×#95—472no—
334C2×#95—472no—
335C2×#95—472no—
336C2×#95—472no—
337C47C2×#232311yescharacteristic, complemented, Sylow
338C25—471no—
339C22×Q8—24no—
340C22×Q8—24no—
341C22×Q8C4×D4718yes—
342C22×Q8C22×D4714yes—
343C23×C4—472no—
344C23×C4—471no—
345C23×C4—478no—
346C23×C4—474no—
347C23×C4—471no—
348C23×C4—471no—
349C23×C4—472no—
350C23×C4—944no—
351C23×C4—944no—
352C23×C4C22×D4712yes—
353C23×C4C22×D4712yes—
354C23×C4C22×D4711yescharacteristic
355C23×C4C22×D4711yescharacteristic
356C23×C4C22×D4711yescharacteristic
357C22⋊Q8—4764no—
358C4×Q8—232no—
359#117—9432no—
360C2×C4⋊C4—28no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—4716no—
363C2×C4⋊C4—478no—
364C2×C4⋊C4—478no—
365C2×C4⋊C4—474no—
366C2×C4⋊C4—474no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—474no—
369C2×C4⋊C4—478no—
370C2×C4⋊C4—944no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—944no—
373C2×C4⋊C4—1884no—
374C2×C4⋊C4C22×D4714yes—
375C2×C4⋊C4C22×D4718yes—
376C2×C22⋊C4—4716no—
377C2×C22⋊C4—478no—
378C2×C22⋊C4—474no—
379C2×C22⋊C4—474no—
380C2×C22⋊C4—478no—
381C2×C22⋊C4—944no—
382C2×C22⋊C4—944no—
383C2×C22⋊C4—1884no—
384C2×C42—28no—
385C2×C42—474no—
386C2×C42—478no—
387C2×C42—944no—
388C2×C42—1884no—
389C2×C42C22⋊D4718yes—
390C2×C42C22×D4718yes—
391C2×C42C22×D4714yes—
392#95—478no—
393#95—478no—
394#95—478no—
395#95—478no—
396#95—478no—
397C24—474no—
398C24—472no—
399C24—942no—
400C24—942no—
401C24—942no—
402C24—942no—
403C24—942no—
404C24—942no—
405C24C23×D4711yescharacteristic
406C2×Q8—232no—
407C2×Q8—28no—
408C2×Q8—28no—
409C2×Q8C2×C4×D47116yes—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—478no—
420C22×C4—474no—
421C22×C4—474no—
422C22×C4—472no—
423C22×C4—4716no—
424C22×C4—472no—
425C22×C4—474no—
426C22×C4—472no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—944no—
430C22×C4—944no—
431C22×C4—9432no—
432C22×C4—948no—
433C22×C4—948no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—944no—
437C22×C4—944no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—944no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4C2×C4×D4718yes—
444C22×C4C2×C4×D4714yes—
445C22×C4#1062535014yes—
446C22×C4C2×C22⋊D4714yes—
447C22×C4C2×C22⋊D4714yes—
448C22×C4C2×C22⋊D4714yes—
449C22×C4C23×D4714yes—
450C22×C4C23×D4714yes—
451C22×C4C23×D4712yes—
452C22×C4C23×D4712yes—
453C22×C4C23×D4712yes—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—4732no—
457C4⋊C4—4716no—
458C4⋊C4—948no—
459C4⋊C4—948no—
460C4⋊C4—948no—
461C4⋊C4—1888no—
462C22⋊C4—4732no—
463C22⋊C4—948no—
464C22⋊C4—948no—
465C22⋊C4—1888no—
466C42—216no—
467C42—948no—
468C42—1888no—
469C42C2×C22⋊D47116yes—
470C23—474no—
471C23—948no—
472C23—948no—
473C23—948no—
474C23—942no—
475C23—942no—
476C23—942no—
477C23—942no—
478C23—942no—
479C23—942no—
480C23—1884no—
481C23—1884no—
482C23—1884no—
483C23—1884no—
484C23C22×C4×D4712yes—
485C23C22×C22⋊D4712yes—
486C23C2×#1062535012yes—
487C23#1309407012yes—
488C23#1309407112yes—
489C23#1309407212yes—
490C23#1309407212yes—
491C23C24×D4711yescharacteristic
492Q8—232no—
493C2×C4—216no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—48no—
506C2×C4—478no—
507C2×C4—474no—
508C2×C4—948no—
509C2×C4—9416no—
510C2×C4—942no—
511C2×C4—942no—
512C2×C4—944no—
513C2×C4—944no—
514C2×C4—9432no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—944no—
518C2×C4—944no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—944no—
524C2×C4—1888no—
525C2×C4—1884no—
526C2×C4—1884no—
527C2×C4C4×C22⋊D47116yes—
528C2×C4C22×C4×D4718yes—
529C2×C4C22×C4×D4714yes—
530C2×C4C22×C22⋊D4714yes—
531C2×C4C22×C22⋊D4714yes—
532C2×C4C22×C22⋊D4714yes—
533C2×C4C2×#1062535014yes—
534C22—944no—
535C22—944no—
536C22—944no—
537C22—18816no—
538C22—1882no—
539C22—1882no—
540C22—1882no—
541C22—1882no—
542C22C2×C4×C22⋊D4714yes—
543C22#1568959914yes—
544C22#1568960514yes—
545C22#1568969714yes—
546C22#1568970114yes—
547C22#1568973114yes—
548C22#1568974514yes—
549C22C23×C4×D4711yescharacteristic
550C22C23×C22⋊D4711yescharacteristic
551C22C22×#1062535011yescharacteristic
552C22C2×#1309407011yescharacteristic
553C22C2×#1309407111yescharacteristic
554C22C2×#1309407211yescharacteristic
555C22C2×#1309407211yescharacteristic
556C4—28no—
557C4—24no—
558C4—24no—
559C4—44no—
560C4—944no—
561C4—948no—
562C4—1882no—
563C4—1882no—
564C4—1884no—
565C4C2×C4×C22⋊D4718yes—
566C2—1884no—
567C2#2024606418yesdirect product factor
568C2C22×C4×C22⋊D4711yescharacteristic
569C2C2×#1568959911yescharacteristic
570C2C2×#1568960511yescharacteristic
571C2C2×#1568969711yescharacteristic
572C2C2×#1568970111yescharacteristic
573C2C2×#1568973111yescharacteristic
574C2C2×#1568974511yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC22×C4×Q8×C47C2
socleC24×C47C24
prime radicalC24×D47C23
cosocleC23×C47C25
hypercenterC22×C4×Q8D47
hypocenterC47C2×#2323
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20246064C2direct (N × H)
4. C2×#15689030C2semidirect
5. C2×#15689019C2semidirect
6. C2×#15689018C2semidirect
9. C2×#15689007C2semidirect
11. C2×C4×C4⋊Dic47C2semidirect
12. C22×C4×Q8×C47C2semidirect
14. C2×#13094000C4semidirect
30. #15689030C22semidirect
31. #15689019C22semidirect
32. #15689018C22semidirect
35. #15689007C22semidirect
37. C4×C4⋊Dic47C22semidirect
45. C2×C4×Q8×C47C22semidirect
60. #13094000C2×C4semidirect
280. C22×C4×Q8D47semidirect
291. C2×C47#2323semidirect
305. C2×C4×Q8C2×D47semidirect
337. C47C2×#2323semidirect
567. C2#20246064direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#130940002
C22×#130939812
#202460642
C2×#156890302
C2×#156890192
C2×#156890182
C2×#156890142
C2×#156890132
C2×#156890072
C2×#156889802
C2×C4×C4⋊Dic472
C22×C4×Q8×C472
C2×#232347